Probability Roadmap Teacher Guide Random Variable Roadmap
Teacher Intervention Guide • HS Statistics
Tier 2 Intervention
Standard: CO HS.S-MD.A.3
Lesson Objective
Students will develop a probability distribution for a discrete random variable by defining the sample space (using tree diagrams) and calculate the expected value \(E(X)\) as a weighted average.
Instructional Scaffolding
Phase 1: The Sample Space Map (10 mins)
Focus on organized counting . Many students struggle to identify all outcomes. Use tree diagrams for sequential events (e.g., flipping 3 coins).
Teacher Action: Model a 2-coin flip. Show how branches represent "choices."Scaffold: Provide "skeleton" tree diagrams where students fill in the outcomes.
Phase 2: Defining the Variable (10 mins)
Bridge the gap between "outcomes" and "numbers." Define \(X\) clearly (e.g., \(X = \text{number of Heads}\)).
Concept Check: Ask, "If I flip T-H-T, what is the value of \(X\)?"Strategy: Use highlighters to group outcomes in the tree diagram that result in the same \(X\) value.
Phase 3: The Distribution Table (10 mins)
Translate the counts into probabilities. Verify that \(\sum P(x) = 1\).
Verification Step: Always ask students to sum their probabilities before moving to expected value.
Phase 4: Expected Value Calculation (15 mins)
Introduce \(E(X) = \sum x \cdot P(x)\) as a "weighted average" or "long-run balance point."
Analogy: Use a seesaw. If more probability is on the higher numbers, the "balance point" shifts right.Calculation Scaffold: Use a 3-column table: \(x\), \(P(x)\), and \(x \cdot P(x)\).
Watch Out For...
The "Average" Trap
Students often just average the \(x\) values (e.g., \((0+1+2)/3\)) instead of weighting them by probability.
Missing Outcomes
In a 3-coin flip, students often forget H-T-H is different from H-H-T. Emphasize the tree diagram branches.
Progress Monitoring Checklist
Can the student draw a complete tree diagram for 3 independent events?
Can the student identify which outcomes correspond to a specific value of \(X\)?
Does the student recognize that the sum of \(P(x)\) must be exactly 1?
Can the student perform the multiplication and summation for \(E(X)\) accurately?
Distribution Discovery Slides Random Variable Roadmap
Predicting the Future with Math
Today's Mission
Map the Sample Space
Find every possible outcome using tree diagrams.
Build a Distribution
Turn outcomes into a table of probabilities.
Find Expected Value
Predict the "average" outcome in the long run.
The Power of Trees
Flipping 2 Coins:
Start at the beginning
Each "fork" is a choice/event
Follow the path to see the result
H T H → HH T → HT H → TH T → TT
What is \(X\)?
A Random Variable (\(X\)) is just a rule that turns an outcome into a number.
The Outcome
H - T - H
The Value (\(X\))
If \(X = \text{total Heads}\), then:
\(X = 2\)
The Distribution Table
\(x\) (The Value) \(P(x)\) (Probability) 0 Heads \(1/4\) 1 Head \(2/4\) 2 Heads \(1/4\) Total Sum 1.0 (or 4/4)
Expected Value
\[E(X) = \sum x \cdot P(x)\]
Multiply → Sum It Up
"On average, what value do we expect if we play the game 1,000 times?"
Value Venture Activity Value Venture
Probability Distribution Workshop
Name:
Date:
1
Challenge: The Three-Coin Toss
You flip a fair coin 3 times. Let \(X\) = the total number of Tails.
Step A: Complete the Tree Diagram
Trace every path to find the total outcomes.
H T [Complete the 3rd set of branches and list outcomes here]
Step B: Create the Table
\(x\) (Heads) \(P(x)\) 0 1 2 3
Step C: Calculate \(E(X)\)
Formula: \(\sum x \cdot P(x)\)
2
Challenge: The Mystery Prize Box
A box contains 5 cards: Two $1 cards, Two $5 cards, and One $20 card. You draw one card at random. Let \(X\) = the value of the card.
1. List the Sample Space
2. Probability Distribution
Value (\(x\)) Probability \(P(x)\) $1 $5 $20
3. Find the Expected Value
Show your multiplication for each row:
Row 1: 1 \(\cdot\) _________ = _________
Row 2: 5 \(\cdot\) _________ = _________
Row 3: 20 \(\cdot\) _________ = _________
Total \(E(X)\) = _________
Thinking Question:
If it costs $8.00 to play this game, would you expect to win or lose money in the long run? Why?
Distribution Check Exit Ticket Exit Ticket: Distribution Check
Progress Monitoring
Name
Score
1
You roll a single 6-sided die. Let \(X\) be the outcome of the roll. Create the probability distribution table below.
Value \(x\) 1 2 3 4 5 6 \(P(X=x)\)
2
Calculate the Expected Value \(E(X)\) for the die roll above. Show your setup.
\(E(X)\) =
3
In your own words: Why do we call it the "Expected" value if we can't actually roll that specific number on the die?
Teacher Feedback
MASTERED
PROGRESSING
NEEDS SUPPORT
Probability Intervention Answer Key Answer Key & Solutions
Random Variable Roadmap Intervention
Value Venture Challenge 1: Three-Coin Toss
Sample Space & Counts
HHH (0 Tails)
HHT, HTH, THH (1 Tail)
TTH, THT, HTT (2 Tails)
TTT (3 Tails)
Total Outcomes: 8
The Distribution Table
\(x\) \(P(x)\) 0 \(1/8\) (0.125) 1 \(3/8\) (0.375) 2 \(3/8\) (0.375) 3 \(1/8\) (0.125)
Expected Value Calculation:
\(E(X) = (0 \cdot 1/8) + (1 \cdot 3/8) + (2 \cdot 3/8) + (3 \cdot 1/8) = 0 + 3/8 + 6/8 + 3/8 = 12/8 = \mathbf{1.5}\) Tails.
Value Venture Challenge 2: Mystery Prize Box
Probability Distribution
Value (\(x\)) \(P(x)\) $1 \(2/5\) (0.4) $5 \(2/5\) (0.4) $20 \(1/5\) (0.2)
Expected Value Setup
Row 1: \(1 \cdot 0.4 = 0.40\)
Row 2: \(5 \cdot 0.4 = 2.00\)
Row 3: \(20 \cdot 0.2 = 4.00\)
Total \(E(X) = \mathbf{\$6.40}\)
Thinking Question Answer:
If it costs $8.00 to play, you would expect to lose money . Since the average payout is only $6.40, you are losing $1.60 per play on average in the long run.
Exit Ticket Solutions
1. Table: All probabilities are \(1/6\).2. \(E(X)\): \((1 \cdot 1/6) + (2 \cdot 1/6) + (3 \cdot 1/6) + (4 \cdot 1/6) + (5 \cdot 1/6) + (6 \cdot 1/6) = 21/6 = \mathbf{3.5}\).3. Concept: "Expected Value" represents the theoretical long-term average if we repeated the experiment many times. It doesn't mean we "expect" to see 3.5 on any single roll, but rather that the average of millions of rolls will settle at 3.5. 
